MAT 105 Fall 2008
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Harder than you might think

There are many examples in history where the
results were disputed

We care about this because we want the
outcome of the election to be “fair”

One purpose of primary elections is to narrow such a
wide field down to a single nominee for each party

A common criticism of the primary process is that
some states are more important than others

Pennsylvania’s primary wasn’t until April 22nd, but the
first was the Iowa caucus on January 3rd

Since the Democratic primary election was so close,
many states with late primaries that ordinarily
wouldn’t have been important were suddenly vital to
each candidate’s success

Whenever an election is
close, there is usually
controversy

The most recent example
of this was the Democratic
primary involving Barack
Obama and Hillary Clinton

Some Clinton supporters still don’t support
Obama, even though they agree on many policies

This is far from the first time there was a
controversial election that got national
attention

We will look at several examples from recent
(and not so recent) years

Al Gore vs. George W. Bush

There were two other candidates:
Ralph Nader (Green Party) and Pat
Buchanan (Reform Party)

The result of the popular vote was:
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Gore 48.4%
Bush 47.9%
Nader 2.7%
Buchanan 0.4%

Even though Al Gore won the popular vote,
George W. Bush won the electoral college

The result of Florida was in dispute for several
weeks, but eventually Florida’s electoral votes
were given to Bush after the Supreme Court
ordered a stop to recounts

The official margin of victory for Bush in Florida
was 537 votes, out of 5.8 million votes cast

The election in 2000 wasn’t the first
time there had been a third-party
“spoiler” in a national election

In 1912, after Theodore Roosevelt
failed to get the Republican party
nomination, he ran as a third-party
candidate

This split the Republican vote, and
Woodrow Wilson, the Democrat,
won the election with only 42% of
the popular vote
Woodrow
Wilson
Theodore
Roosevelt
William
Howard Taft

The example from 1912 teaches us that two
candidates can “split” a large portion of the
voters, leading to an unanticipated outcome

For example, if Mitt Romney had decided,
after losing the Republican nomination to
John McCain, to run as a third-party
candidate, this would almost certainly
guarantee that the Democrats would win the
2008 Presidential race

Three main candidates:
 Norm Coleman (R)
 Hubert Humphrey (D)
 Jesse Ventura (Reform Party)

Norm
Coleman
The results of the vote were:
 Ventura (37%)
 Coleman (34%)
Hubert
Humphrey
 Humphrey (28%)
Jesse
Ventura

Few people expected the former
professional wrestler to win the
election

Most of the people who voted for
Coleman or Humphrey probably had
Ventura as their last choice

That means that Ventura was elected
governor even though 63% of the
voters would have ranked him last!
There are many different systems, as we will learn
 The most common system used in US elections is the
plurality system: the candidate who gets more votes than
any other candidate is said to receive a “plurality”
 A candidate receives a “majority” if they earn more than
half of the total number of votes

 Al Gore won a plurality of the popular vote in 2000
 Woodrow Wilson won a plurality of the popular vote in 1912
 Jesse Ventura won a plurality of the vote in 1998

None of these candidates won a majority

In most US elections, voters can only cast a single
ballot for the candidate he or she likes the best

However, most voters will have “preference
lists”: a ranking of the candidates in order of
most preferred to least preferred

For example, many (but not all) of the people
who voted for Ralph Nader in 2000 would have
had Al Gore as their second choice

Suppose a class of children is trying to decide what
drink to have with their lunch

The choices are milk, soda, and juice

Each child votes for their top choice

The results are:
 Milk 6
 Soda 5
 Juice 4

Milk wins a plurality of the votes, but not a majority

What if we ask the children to rank the drinks
in order of preference?
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6 have the preference Milk > Soda > Juice
5 have the preference Soda > Juice > Milk
4 have the preference Juice > Soda > Milk

Is the outcome fair? If we choose Milk as the
winner of this election, 9 of the 15 students
are “stuck” with their last choice

We will not allow ties on individual preference
lists, though some methods will result in an
overall tie

All candidates must be listed in a specific order

We will sometimes assume that the number of
voters is odd to avoid ties (remember we will
think about applying these methods to situations
where we have thousands or millions of voters)
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We’ll start off simple and only consider the case
where we have two candidates

There are only two preferences: A > B and B > A
 Voters with preference A > B vote for A
 Voters with preference B > A vote for B

The candidate with the most votes wins

This method is called majority rule

Notice that one of the two candidates will
definitely get a majority (they can’t both get
less than half of the votes)

Majority rule has three desirable properties
 anonymous
 neutral
 monotone

If any two voters exchange (marked) ballots
before submitting them, the outcome of the
election does not change

In this way, who is casting the vote doesn’t
impact the result of the vote; all the voters
are treated equally

If a new election were held and every voter
reversed their vote (people who voted for A
now vote for B, and vice versa), then the
outcome of the election is also reversed

In this way, one candidate isn’t being given
preference over another; the candidates are
treated equally

If a new election were held and a single voter
were to change his or her ballot from being a
vote for the loser of the previous election to
being a vote for the winner of the previous
election, and everyone else voted exactly as
before, then the outcome of the new election
would be the same as the outcome of the
previous election

Changing your vote from the loser to the winner
shouldn’t help the loser

Majority rule is not the only way to determine
the winner of an election with two candidates

May’s Theorem states that majority rule is
the only method for determining the winner
of an election with two candidates that
satisfies all three conditions: anonymous,
neutral, and monotone
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Matriarchy: only the votes of women count
Dictatorship: there is a certain voter called the
dictator, and only the dictator’s vote counts (all other
ballots are ignored)
Oligarchy: there is a small council of voters, and only
their votes count (think of the oligarchs as “codictators”)
Minority rule: the candidate who gets the fewest
votes wins
Imposed rule: a certain candidate wins no matter
what the votes are

All of these methods are “unfair,” but fairness
can be a very subjective concept

May’s Theorem gives us a way to think of
fairness objectively

An election method that satisfies all three
conditions is “fair,” and a method that does
not isn’t